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Mirrors > Home > MPE Home > Th. List > oppgsubg | Structured version Visualization version GIF version |
Description: Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
Ref | Expression |
---|---|
oppggic.o | β’ π = (oppgβπΊ) |
Ref | Expression |
---|---|
oppgsubg | β’ (SubGrpβπΊ) = (SubGrpβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgrcl 18856 | . . 3 β’ (π₯ β (SubGrpβπΊ) β πΊ β Grp) | |
2 | subgrcl 18856 | . . . 4 β’ (π₯ β (SubGrpβπ) β π β Grp) | |
3 | oppggic.o | . . . . 5 β’ π = (oppgβπΊ) | |
4 | 3 | oppggrpb 19061 | . . . 4 β’ (πΊ β Grp β π β Grp) |
5 | 2, 4 | sylibr 233 | . . 3 β’ (π₯ β (SubGrpβπ) β πΊ β Grp) |
6 | 3 | oppgsubm 19065 | . . . . . . 7 β’ (SubMndβπΊ) = (SubMndβπ) |
7 | 6 | eleq2i 2828 | . . . . . 6 β’ (π₯ β (SubMndβπΊ) β π₯ β (SubMndβπ)) |
8 | 7 | a1i 11 | . . . . 5 β’ (πΊ β Grp β (π₯ β (SubMndβπΊ) β π₯ β (SubMndβπ))) |
9 | eqid 2736 | . . . . . . . . 9 β’ (invgβπΊ) = (invgβπΊ) | |
10 | 3, 9 | oppginv 19062 | . . . . . . . 8 β’ (πΊ β Grp β (invgβπΊ) = (invgβπ)) |
11 | 10 | fveq1d 6827 | . . . . . . 7 β’ (πΊ β Grp β ((invgβπΊ)βπ¦) = ((invgβπ)βπ¦)) |
12 | 11 | eleq1d 2821 | . . . . . 6 β’ (πΊ β Grp β (((invgβπΊ)βπ¦) β π₯ β ((invgβπ)βπ¦) β π₯)) |
13 | 12 | ralbidv 3170 | . . . . 5 β’ (πΊ β Grp β (βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯ β βπ¦ β π₯ ((invgβπ)βπ¦) β π₯)) |
14 | 8, 13 | anbi12d 631 | . . . 4 β’ (πΊ β Grp β ((π₯ β (SubMndβπΊ) β§ βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
15 | 9 | issubg3 18869 | . . . 4 β’ (πΊ β Grp β (π₯ β (SubGrpβπΊ) β (π₯ β (SubMndβπΊ) β§ βπ¦ β π₯ ((invgβπΊ)βπ¦) β π₯))) |
16 | eqid 2736 | . . . . . 6 β’ (invgβπ) = (invgβπ) | |
17 | 16 | issubg3 18869 | . . . . 5 β’ (π β Grp β (π₯ β (SubGrpβπ) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
18 | 4, 17 | sylbi 216 | . . . 4 β’ (πΊ β Grp β (π₯ β (SubGrpβπ) β (π₯ β (SubMndβπ) β§ βπ¦ β π₯ ((invgβπ)βπ¦) β π₯))) |
19 | 14, 15, 18 | 3bitr4d 310 | . . 3 β’ (πΊ β Grp β (π₯ β (SubGrpβπΊ) β π₯ β (SubGrpβπ))) |
20 | 1, 5, 19 | pm5.21nii 379 | . 2 β’ (π₯ β (SubGrpβπΊ) β π₯ β (SubGrpβπ)) |
21 | 20 | eqriv 2733 | 1 β’ (SubGrpβπΊ) = (SubGrpβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1540 β wcel 2105 βwral 3061 βcfv 6479 SubMndcsubmnd 18526 Grpcgrp 18673 invgcminusg 18674 SubGrpcsubg 18845 oppgcoppg 19045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-tpos 8112 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-grp 18676 df-minusg 18677 df-subg 18848 df-oppg 19046 |
This theorem is referenced by: lsmmod2 19377 lsmdisj2r 19386 |
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